A tale of two vortices: How numerical ergodic theory and transfer operators reveal fundamental changes to coherent structures in non-autonomous dynamical systems

Figure 1.  Figures 1a and 1b show almost invariant structures as described by the (evolved) subdominant eigenvector of an Ulam matrix approximation to the transfer operator in the periodically driven double gyre flow, with parameters as in [40]. Figures 1c and 1d show finite-time coherent structures as described by the (evolved) subdominant initial time singular vector of a composition of $ 10 $ Ulam matrices describing the evolution of the transitory double gyre flow introduced by [31]. See [17] for a thorough discussion of both models

Figure 2.  Evolution in non-autonomous dynamical systems: driving system (above the arrow), particle evolution (2nd row), transfer operators (3rd row) and Ulam's method (bottom row)

Figure 3.  Selected vector field instances for the periodically forced double well potential

Figure 10.  An illustration of the behaviour of $ \alpha(t) $ and $ \tilde{\alpha}(t) $ over $ 5 $ periods

Figure 4.  Tracking modes over rolling windows for the periodically forced double well potential

Figure 5.  Tracking modes for time windows of length $ n = 50 $, evolved using Algorithm 4

Figure 6.  Crossing introduced by shifting from $ n = 54 $ to $ n = 51 $ for the periodically forced double well potential

Figure 7.  Equivariance mismatch for the periodically forced double well potential when $ n = 50 $

Figure 8.  Leading $ 6 $ of $ \mathcal{N} = 6 $ modes for the periodically forced double well potential when $ n = 50 $

Figure 9.  Leading $ 4 $ of $ \mathcal{N} = 4 $ modes for the periodically forced double well potential when $ n = 100 $

Figure 11.  Mean equivariance mismatch, as per Algorithm 5, for the leading $ 4 $ of $ \mathcal{N} $ modes using the two pairing methods given by Algorithms 2 ($ \bar{\varsigma}_{S} $) and 3 ($ \bar{\varsigma}_{U} $) for $ n = 50 $

Figure 12.  Leading $ 4 $ from a total $ \mathcal{N} = 5 $ tracked modes for $ n = 50 $ using Algorithms 3 (top) and 5 (bottom)

Figure 13.  Leading $ 4 $ from a total $ \mathcal{N} = 7 $ tracked modes for $ n = 50 $ using Algorithms 3 (top) and 5 (bottom)

Figure 14.  Consecutive windows corresponding to reasonable equivariance for $ S_{U}^{(4)} $ of Figure 12

Figure 15.  Initial time singular vectors corresponding to rolling windows initialised at the various $ t_{0} $ indicated by colour coded bars and column headings. These are paired according to the paths illustrated in Figure 12

Figure 16.  Evolved $ u^{(50)}_{75,4} $ (top) and $ u^{(50)}_{274,4} $ (bottom) of mode $ S_{U}^{(4)} $ in Figure 15, evolved as per Algorithm 4

Figure 17.  Southern hemisphere wind speed (easterly and northerly) on the $ 850 $ K isentropic surface

Figure 18.  Mean equivariance mismatch, as per Algorithm 5, for the leading $ 3 $ of $ \mathcal{N} $ modes using the two pairing methods given in Algorithms 2 and 3 with $ n = 56 $ and $ t_0 \in[0000 \: 1 \: \text{August}, 1800 \: 30 \: \text{September}] $. Here the Ulam matrices, describing transitions for the area south of $ 30^{\circ} $S, are of dimension $ m \times m $ for $ m = 2^{14} $

Figure 19.  Leading $ 3 $ of $ \mathcal{N} = 3 $ tracked paths of singular values of rolling windows paired using Algorithm 2 for $ n = 56 $

Figure 20.  Leading singular vectors, for various $ t_{0} $, of matrix compositions associated with Figure 19b where time windows are of length $ n = 56 $. The area illustrated is south of $ 50^{\circ} $S and the time given in the label is the relevant $ t_{0} $ for that window

Figure 21.  Evolved leading mode associated with Figure 19a for a time window centred on the peak at $ 1800 $ on $ 23 $ Sep. This is illustrated on the area south of $ 15^{\circ} $S

Figure 22.  Evolved subdominant mode associated with Figure 19b for a time window centred on the peak at $ 0600 $ on $ 24 $ Sep. This is illustrated on the area south of $ 15^{\circ} $S

Figure 23.  Evolved leading singular vectors for time windows centred at $ 0000 $ on $ 24 $ Sep. for $ m = 12,800 $ initially seeded bins whose centres are south of $ 20^{\circ} $S. This is illustrated on the full southern hemisphere

Figure 24.  Evolved subdominant mode normalised as in [19] for time windows centred at $ 0000 $ on $ 24 $ Sep. for $ m = 12,800 $ initially seeded bins whose centres are south of $ 20^{\circ} $S. This is illustrated on the full southern hemisphere